I just came across this approximation that I have no idea how it was derived:
$e^{i\beta sin(\Omega t)} \approx J_0(\beta) + 2iJ_1(\beta)sin(\Omega t)$
I thought of using Moivre theorem and the fact that
$cos(\beta sin(\Omega t)) = \sum_{m=-\infty}^{m=\infty}J_mcos(m \Omega t) $
$sin(\beta sin(\Omega t)) = \sum_{m=-\infty}^{m=\infty}J_msin(m \Omega t) $
Which gets me close to the approximation but I don't know how to approximate and get rid of the summation or if I'm in the right track. Can anyone help me? The paper just pulled this approximation out of thin air and I can't find anything related to it on the web. Thanks.